For naturally fractured reservoirs, it is very difficult to quantify future prediction without proper fracture properties, including such factors as fracture permeability, fracture orientation, and fracture length, and capillary pressure. Capillary pressure and fracture permeability can interact in a way that can produce nonunique results. Evaluation of core data may provide us with some information about imbibition capillary pressure curves; however, quantification of fracture permeability is determined largely by matching production data.
This paper presents an integrated approach to history matching naturally fractured reservoirs by adjusting the fracture permeability of individual fractures and water/oil capillary pressure curves. This paper is focused mainly on history-matching procedure and not on geological-model construction. By minimizing an objective function to history matching production data, we generate estimates of fracture permeability of individual fractures and water/oil capillary pressure curves. All implementations were incorporated into a commercial simulator and iterated in the automatic-history-matching scheme. The adjoint method and an efficient direct solver were used to reduce central-processing-unit (CPU) time for calculating the sensitivity-coefficient matrix. A 2D synthetic case was used, with the fracture distribution from a Middle East reservoir, to validate this method.
Naturally fractured reservoirs have received much more attention in the last decade because of low oil recovery in many of them. Fluid flow in naturally fractured reservoirs is controlled largely by the distribution, orientation, and interconnectivity of the fracture system (Bourbiaux et al. 2005). A great deal of effort has been made in characterization of the geological fracture-network system based on the analysis, interpolation, and extrapolation of the fracture information acquired in wells, derived from seismic data, and obtained from outcrop analog data. As a result, there has been a large amount of work concerned with diagnosing and characterizing fractured reservoirs and developing viable techniques to use fractures effectively (Barenblatt et al. 1960; Bourbiaux et al. 1998, 2002; Dershowitz et al. 2000; Hu and Jenni 2005; Thomas et al. 1983, 1991; Gilman and Kazemi 1983). The reservoir characterization, modeling, and simulation of naturally fractured reservoirs present unique challenges compared with conventional reservoirs, because the fracture, the matrix, and the mass transfer between matrix and fracture have to be characterized properly. In this paper, production data were incorporated into the static model to characterize the fracture permeability of individual fractures, which is difficult to quantify properly without production data.
The capillary pressure in naturally fractured reservoirs plays a much more important role than that in conventional reservoirs. Capillary forces in fractured reservoirs are an important component of driving mechanisms, while the dynamic role of the capillary forces in a conventional reservoir is much more limited. In a fractured reservoir, capillary forces may contribute to the displacement process through imbibition, or they may oppose it through drainage (Reiss 1981).
It is essential to represent capillary pressure properly. Generally there are four different methods of measuring capillary pressure curves, including porous plate, centrifuge, air mercury injection, and water-vapor desorption. These methods are based on the availability of core data and suffer limitation because of scales over which the data are collected. It would be very useful if capillary pressure curves could be extracted from production data through the history-matching process.
Generally, geological models for naturally fractured reservoirs derived from static data alone cannot reproduce the field production history, which might be ascribed to the insufficient consideration of fracture effects on flow, insufficient dynamic characterization of the distribution of fracture systems, and insufficient consideration of the interaction between matrix and fracture. In other words, the hydrodynamic properties of a fracture-network system and capillary pressure curves need to be characterized properly. In this paper, production data were used to characterize fracture permeability and to estimate water/oil capillary pressure curves, which mainly control hydrodynamic properties of naturally fractured reservoirs.
Reconciling geologic models to the dynamic response of the reservoir is critical to building reliable reservoir models, which are very important in reservoir management and key economic decision making. Classical history-matching procedures whereby the reservoir parameters are adjusted manually by trial-and-error procedure can be tedious and time consuming. In recent years, several techniques have been developed for integrating production data to improve reservoir models, but only a few of these papers have been focused on naturally fractured reservoirs (Oliver 1994; Gao and Reynolds 2006; Cheng et al. 2004; Landa and Horne 1997; Cui and Kelkar 2005; Gang 2006; Gang and Kelkar 2006). The history matching of naturally fractured reservoirs requires that both the matrix and the fracture system be quantified properly. The field experience suggests that it is very difficult to quantify the fracture system properly without production data, while the matrix system might be characterized reasonably with static data. No research work so far has focused on estimating capillary pressure curves and fracture permeability by use of production data.
The history-matching process generally requires a optimization algorithm to find the minimum of an objective function that defines the mismatch between the observed and calculated data. Gradient-based algorithms are used widely to minimize the objective function in automatic history matching. For this algorithm, with an initial estimate of model parameters--in this case, the fracture permeability and water/oil capillary pressure parameters--the optimization algorithm will calculate the change of the model parameters with information from the computation of gradient and the Hessian of the objective function. The sensitivity-coefficient matrix, the partial derivatives of production data with respect to model parameters, is needed to evaluate the gradient and the Hessian.
In this paper we propose an approach to history matching fractured reservoirs through adjusting the fracture permeability of individual fractures and the capillary pressure parameters. First, with the initial estimate of model parameters, the adjoint method was used to calculate the sensitivity of production data with respect to model parameters, grid permeability, and capillary pressure. An efficient direct solver was used to reduce the CPU time, considering the special linear-algebra problem in this process. Second, the sensitivity of production data with respect to fracture permeability was computed through the chain rule. Third, changes of fracture permeability and capillary pressure parameters were computed by use of an optimization algorithm with information from the computation of the gradient and the Hessian. Fourth, the fracture permeability and capillary pressure curves were updated, and then the grid permeability was updated with new fracture permeability. Assuming that the relation between fracture permeability and grid permeability is known is a limitation of this paper. Though previous researchers have made assumptions regarding this relationship (da Silva 1989; Thomas et al. 1991; Cui and Kelkar 2005), we cannot confirm its validity independently. Fifth, the updated model was simulated to predict the reservoir response. The above process was repeated until satisfactory history-matching results were obtained. Fig. 1 shows the flow chart of this history-matching process. This approach avoids the time-consuming trial-and-error procedure associated with manual history matching. Through adjusting the fracture permeability instead of grid permeability directly, the geological consistency is preserved. The estimation of capillary pressure curves is very useful for quantifying reservoir performance when limited core data are available in naturally fractured reservoirs.